triangle matrix respectively. For operation U(union) and n(intersection), there are many properties, uch as commutative law, associative law, idempotent law, and eve distributive law etc In previous example, we have found many properties. However, a lattice generally have the following common properties Proposition 7. The Lattice with operation n and U satisfies 1. Commutative:anb=b∩a,aUb=b∪a. 2. Associative:(anbnc=an(bnc),(aUbUc=aU(bUc) 3. Idempotent: ana=a, aUa=a 4. Absorption:(aUb)na=a,(anhua=a Proof. The proof is directly deduced from definition, which is left as an exercise 3.2 Algebraic Perspective In abstract algebra, we first learn semigroup. Then more and more complicated structure are introduced by adding more constraints. Similarly, We also introduce semilattice here. Definition 8. A semilattice is an algebra S=(S, *)satisfying, for all a, y, z ES, 2.x*y=y* 9.x*(y*2)=(x*y)* Example 4. Given a set A. Consider the partially ordered set((A), S). Then(P(A),U semilattice For this case, it is easy to verify that all three properties are satisfied Based on semilattice, we can furthermore define lattice by defining two operations n and U on some set as following Definition 9. Given a structure L=(L, n, U), it is a lattice if it subjects to 1.(L,n) and(L, n)are two semilattices 2.(aUb)na=a,(anhUa=a This is a typical style of defining a structure in algebra. In another word, Lattice is just a structure with two operations n and U on some set which meets 4 properties mentioned previously in Proposition 7. Therefore, we have the following theorem to guarantee the equivalence of two different definitions.triangle matrix respectively. For operation ∪(union) and ∩(intersection), there are many properties, such as commutative law, associative law,idempotent law, and eve distributive law etc.. In previous example, we have found many properties. However, a lattice generally have the following common properties. Proposition 7. The Lattice with operation ∩ and ∪ satisfies: 1. Commutative: a ∩ b = b ∩ a, a ∪ b = b ∪ a. 2. Associative: (a ∩ b) ∩ c = a ∩ (b ∩ c),(a ∪ b) ∪ c = a ∪ (b ∪ c). 3. Idempotent: a ∩ a = a, a ∪ a = a. 4. Absorption: (a ∪ b) ∩ a = a,(a ∩ b) ∪ a = a. Proof. The proof is directly deduced from definition, which is left as an exercise. 3.2 Algebraic Perspective In abstract algebra, we first learn semigroup. Then more and more complicated structure are introduced by adding more constraints. Similarly, We also introduce semilattice here. Definition 8. A semilattice is an algebra S = (S, ∗) satisfying, for all x, y, z ∈ S, 1. x ∗ x = x, 2. x ∗ y = y ∗ x, 3. x ∗ (y ∗ z) = (x ∗ y) ∗ z. Example 4. Given a set A. Consider the partially ordered set hP(A), ⊆i. Then hP(A), ∪i is a semilattice. For this case, it is easy to verify that all three properties are satisfied. Based on semilattice, we can furthermore define lattice by defining two operations ∩ and ∪ on some set as following: Definition 9. Given a structure L = (L, ∩, ∪), it is a lattice if it subjects to: 1. (L, ∩) and (L, ∩) are two semilattices. 2. (a ∪ b) ∩ a = a,(a ∩ b) ∪ a = a. This is a typical style of defining a structure in algebra. In another word, Lattice is just a structure with two operations ∩ and ∪ on some set which meets 4 properties mentioned previously in Proposition 7. Therefore, we have the following theorem to guarantee the equivalence of two different definitions. 4